Vibration of High-Speed Spur Gear Webs

Abstract

High cycle fatigue loading of gear webs due to in-plane stresses, caused by forced excitation resulting from centrifugal loading and dynamic tooth loads, has been known to cause radial fatigue cracks. This is especially prevalent in high-speed gears used in aerospace applications, with small web thickness, for weight reduction. Radial cracks have also been observed to originate at the outer edge of lightening holes machined in gear webs for weight reduction. This paper presents an analytical treatment of the in-plane vibration of high-speed gear webs resulting from rotational effects and periodic excitation from dynamic tooth loading. Dynamic tooth loads result from the combined effect of inertia forces of gear wheels which are significant at high speeds, the periodic variation of gear mesh stiffness, and involute tooth profile errors. The gear web is modeled as a thin rotating disc and the governing differential equations of motion and the associated boundary conditions are derived from first principles. The equations are then nondimensionalized which leads to some essential nondimensional parameters. A comprehensive tooth stiffness model for spur gears is used that accounts for periodic variation of mesh stiffness. The dynamic tooth loads are obtained by solving the pertinent equations of motion, using a collocation method, that yields a closed-form expression for the periodic excitation, that is used as an input for the in-plane vibration problem. The in-plane vibration equations are solved by an approximate method of weighted residuals. It is found that the displacement fields and the resulting stresses can be significant under certain speeds and loading conditions. The interaction between the forcing frequencies due to gear teeth dynamics and the in-plane vibration natural frequencies can result in resonances that induce high fatigue stresses in the gear web. The in-plane stresses leading to high cycle fatigue loading, and frequency components of the resulting response are discussed in detail.